What percentage of the sky is occluded by stars? If you drew rays from the center of the earth out to infinity at every angle, what percentage of them would intersect a star?
Extra details: 


*

*Assume the rays are mathematical rays, or that they travel at infinite speeds.

*Even in an infinite universe, since the apparent magnitude of stars decreases with distance, it is conceivable that the area occluded is less than 1, i.e. that the total magnitude of the stars at a given distance approaches zero as the distance increases.

*Assume a snapshot of the universe as it currently exists.
 A: With the assumptions you have added to the question the answer would be that every ray would end on a star and that the night sky would be as bright as the surface of the average star - i.e. quite bright.
You are essentially describing Obler's paradox.  Per Wikipedia:

In astrophysics and physical cosmology, Olbers' paradox, named after
  the German astronomer Heinrich Wilhelm Olbers and also called the
  "dark night sky paradox", is the argument that the darkness of the
  night sky conflicts with the assumption of an infinite and eternal
  static universe. The darkness of the night sky is one of the pieces of
  evidence for a non-static universe such as the Big Bang model. If the
  universe is static and populated by an infinite number of stars, any
  sight line from Earth must end at the (very bright) surface of a star,
  so the night sky should be completely bright. This contradicts the
  observed darkness of the night.  (see http://en.wikipedia.org/wiki/Olbers%27_paradox)

With the questions assumptions of a "snapshot" of the universe and that light travels at infinite speed along with the additional assumption that the universe is infinite,the result would be a very bright sky.  Now the measured curvature of the universe is flat to within about 1%, so if the universe is in fact flat or even negatively curved the universe would be infinite and you would obtain Oblers' paradox.  If it turns out that the universe is positively curved it would be finite but you would still have Oblers' paradox since the ray that "circles" the universe without hitting a star would keep circling the universe until it does hits a star.
The inverse square law does not diminish the brightness since the number of stars at a given distance increases as the square of the distance, so the two factors cancel each other out.  
The reasons our night sky is not as bright as the surface of the sun is:


*

*The universe is 13.7 +/- 0.15 billion years old so that light from
stars that are further away than 13.7 billion light years could not
have reached us.

*The universe is expanding so the light from stars and galaxies that
are far away from us will be redshifted and will not be in the
visible part of the light spectrum.

*There is dust and other dark objects that can absorb light from more distant stars.


There is one other reason why the sky should be as bright(er) than a stars surface. Up until 380,000 years after the Big Bang, the entire universe was filled with a very hot plasma. At 380,000 years it had cooled enough for the electrons and protons/nuclei to recombine and produce neutral hydrogen and neutral helium etc. At that time the universe became transparent and was filled with the photons from that the hot ionized plasma.  However since then the continued expansion of the universe has redshifted those visible light photons down into the microwave range. That is the source of the Cosmic Microwave Bacground (CMB) radiation that now has an effective temperature of 2.7 degrees Kelvin. However, there are approximately 1,000,000,000 of those photons per proton in the universe so the sky is still quite bright, just not in photons that our eye can see.
